Applied Mathematics
Level 6
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2 • Applied Mathematics
INTRODUCTION
Hi, I’m EdWIN
Hi, my name is EdWIN. I will be your guide
through Applied Mathematics Level 6. Together we will
proceed through this course at your speed. Look for
me to pop up throughout your lessons to give you
helpful tips, suggestions, and maybe even a pop quiz
question or two. Don’t worry. You can find the answers
to all quiz questions at the end of the course.
Now, don’t get nervous. We will cover one topic at
a time and I will be there with examples to help you
along.
If the content of the lesson is something that you
understand, you should be able to work through it at a
faster pace. On the other hand, if the material is difficult,
read the exercises one at a time. After you try one
problem, look at the solution. You can learn by
reviewing each step that is provided in the solution and
by concentrating on the process being illustrated.
Level 6 of Applied Mathematics is designed to help
workers solve more complex problems than in previous
levels. Many problems involve several steps of reasoning
or may require manipulation of formulas, multiple rate
calculations, comparisons, or conversions. You will be
performing numerical calculations with more
complicated numbers including fractions, mixed
numerals, and negative numbers. Volume of rectangular
solids is introduced. Because workers are often in
positions in which they must rely on a co-worker’s work,
Level 6 teaches how to check answers for correctness
and correct mistakes.
Applied Mathematics • 3
OUTLINE
LESSON 1
Review of Prerequisite Skills
LESSON 2
Review of Fractions
LESSON 3
Introduction to Negative Numbers
LESSON 4
Multiplying and Dividing with Negative
Numbers
LESSON 5
Review of Percent Problems
LESSON 6
Solving Multiple Rate Problems
LESSON 7
Review of Perimeter and Area
LESSON 8
Introduction to Volume
LESSON 9
Applications of Multistep Word Problems
LESSON 10
REFERENCES
4 • Applied Mathematics
Posttest
Workplace Problem Solving Glossary
Test-Taking Tips
Formula Sheet
LESSON 1
REVIEW OF PREREQUISITE SKILLS
Let’s see if you are ready for this level by completing
the pretest. The answers will be provided following the
test. You should be able to complete all of the problems
on the pretest. If you cannot, it would be helpful to
review these skills before you begin this course.
Good luck!
Ready, set, go!
Applied Mathematics • 5
LESSON 1
EXERCISE – PRETEST
Instructions:
Perform the indicated operations.
1.
45 + 68 =
2.
1 3
+ =
5 5
3.
1 1
÷ =
2 4
4.
1 1
× =
2 4
5.
54.890 - 0.00002 =
6.
79.2 ÷ 48 =
7.
7 6
− =
8 8
6 • Applied Mathematics
LESSON 1
8.
What is 78% of 12?
9.
Find the perimeter and area of a square with sides 6 inches in length.
10.
Find the circumference and area of a circle with a diameter of 12
cm. (Use the approximation of 3.14 for π)
11.
Convert 75% to a fraction and a decimal.
12.
Samuel bakes cakes. If he bakes 7 cakes in 2 hours, how many
does he bake in 5 hours?
13.
Reggie is paid by commission and earned $1,575 last month. This
month he made $1,250. What was the percent of decrease in
Reggie’s wages?
Applied Mathematics • 7
LESSON 1
14.
It takes 26 man hours to produce 4 cases of hair brushes. A
company named Split Ends orders 40 cases of shampoo, 4 cases
of hair spray, 16 cases of nail polish, and 10 cases of hair brushes.
They will pick up the shipment, so there are no shipping costs
incurred. How many man hours are necessary to produce the
number of hair brushes ordered?
15.
6.25 lb = ____ oz?
8 • Applied Mathematics
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Applied Mathematics • 9
LESSON 1
ANSWERS TO EXERCISE
1.
45 + 68 =
Answer:
2.
1 3
+ =
5 5
Answer:
3.
1
8
54.890 - 0.00002 =
Answer:
6.
2
1 1
× =
2 4
Answer:
5.
4
5
1 1
÷ =
2 4
Answer:
4.
113
54.88998
79.2 ÷ 48 =
Answer:
10 • Applied Mathematics
1.65
LESSON 1
7.
7 6
− =
8 8
Answer:
8.
What is 78% of 12?
Answer:
9.
Perimeter = 4 sides × 6 in = 24 in
Area = 6 in × 6 in = 36 sq in
Find the circumference and area of a circle with a diameter
of 12 cm. (Use the approximation of 3.14 for π)
Answer:
11.
78% × 12 = 9.36
Find the perimeter and area of a square with sides 6 inches in length.
Answer:
10.
1
8
C = πd
C = 3.14 × 12 = 37.68 cm
A = πr2
A = 3.14 × 62 = 113.04 cm2
Convert 75% to a fraction and a decimal.
Answer:
75% =
75 3
=
100 4
75% = .75
This is a common percent. It will be helpful to memorize.
Applied Mathematics • 11
LESSON 1
12.
Samuel bakes cakes. If he bakes 7 cakes in 2 hours, how many
does he bake in 5 hours?
Answer:
7 x
=
2 5
2x = 35
x = 17.5
17
13.
1
cakes in 5 hours
2
Reggie is paid by commission and earned $1,575 last month. This
month he made $1,250. What was the percent of decrease in
Reggie’s wages?
Answer:
1,575 - 1,250 = 325
325
= .206
1,575
21% decrease
14.
It takes 26 man hours to produce 4 cases of hair brushes. A
company named Split Ends orders 40 cases of shampoo, 4 cases
of hair spray, 16 cases of nail polish, and 10 cases of hair brushes.
They will pick up the shipment, so there are no shipping costs
incurred. How many man hours are necessary to produce the
number of hair brushes ordered?
4 cases
Answer:
26 man hours
65 man hours
12 • Applied Mathematics
=
10 cases
x man hours
LESSON 1
15.
6.25 lb = ____ oz?
Answer:
100 ounces
Check out the suggested
activities on page 13.
Applied Mathematics • 13
LESSON 2
REVIEW OF FRACTIONS
Lesson 2 will begin with a review of multiplying
and dividing fractions. You should already know the
basics. Multiplying and dividing fractions are part of
your prerequisite skills for this course. If you already
know how to do this, you might choose to skip over to
the exercises. If not - here’s a short review.
MULTIPLYING
Multiply straight across, top numbers by top
numbers and bottom numbers by bottom numbers.
You can “cancel” any number or factor that is found in
both the top and bottom of the fractions.
Example:
3 5
× (3 will cancel with 6)
4 6
3/ 1 5
× (3 goes into itself one time and into 6 twice)
4 6/ 2
1× 5 5
=
4×2 8
Example:
2 7
2 7/ 1
2 1 2
× → 1×
→ × =
7 9
7/
9
1 9 9
14 • Applied Mathematics
LESSON 2
DIVIDING
The steps for dividing fractions are the same, except
you must flip the number on the right (called the
divisor) and change the operation to multiplication.
3 5 
5
3 6
÷ →  flip  → × (change to multiplication)
4 6 
6
4 5
3 6/ 3 3 × 3 9
× =
=
(4 and 6 will reduce by 2)
4/ 2 5 2 × 5 10
If the result in either case, multiplication or division,
can be simplified, you should do so. However, if all
factors are canceled before multiplying, the answer will
always be in reduced (simplified) form.
Applied Mathematics • 15
LESSON 2
For example:
6 4
6 3 18
÷ → × =
7 3
7 4 28
(needs reduced since the common 2 in 6 and 4 were not taken out
before the fractions were multiplied)
18 ÷ 2
9
(both top and bottom numbers have a factor of 2) =
28 ÷ 2
14
Always reduce fractions. Remember, you might have the
key
on the calculator. To enter a fraction problem:
3 5
× =
4 6
Press: 3
4×5
6=
5
8
This will give you the reduced answer. Try practicing multiplication
and division of fractions on your own.
16 • Applied Mathematics
LESSON 2
EXERCISE – REVIEW OF MULTIPLYING AND DIVIDING FRACTIONS
Instructions:
1.
2 1
× =
3 4
2.
4 3
× =
9 9
3.
1 4
× =
4 5
4.
3 14
÷
=
4 12
5.
4 3
÷
=
5 10
6.
1 4
÷ =
3 3
Multiply or divide and simplify when possible.
Applied Mathematics • 17
LESSON 2
ANSWERS TO EXERCISE
1.
2 1
× =
3 4
Answer:
2.
4 3
× =
9 9
Answer:
3.
//3
3 14 3 12
3 12
3 ×3 9
÷
= ×
→ 1×
→
=
4 12 4 14
4/
14
1 ×14 14
4 3
÷
=
5 10
Answer:
6.
1 4/ 1
1 ×1 1
×
→
=
1
4/
5
1×5 5
3 14
÷
=
4 12
Answer:
5.
4 3/ 1
4 ×1
4
4 3/ 1 4 ×1
4
or × 3 =
→
=
=
×
3
9/
9
3 × 9 27
9 9/
9 × 3 27
1 4
× =
4 5
Answer:
4.
2/ 1 1
1 ×1 1
× 2 →
=
3 4/
3 ×2 6
/
/ /2
4 3
4 10
4 10
4×2 8
2
÷
→
×
→ 1×
→
= or 2
5 10
5 3
5/
3
1×3 3
3
1 4
÷ =
3 3
Answer:
18 • Applied Mathematics
1 4
1 3
1 3/ 1
1 ×1 1
÷
→
×
→ 1×
→
=
3 3
3 4
3/
4
1×4 4
LESSON 2
I trust you did well on the exercises. The remainder
of this lesson will emphasize addition and subtraction
of fractions. You should already know how to add and
subtract fractions with the same bottom numbers
(denominators). I’ll do a quick review.
1 7
1+ 7
8
1
carry the denominator =
reduce by 8 = = 1
+
→
8 8
8
8
1
(
)
(
)
Subtraction works the same way:
7 1
7−1 6
−
→
=
8 8
8
8
(reduce by 2) = 34
Fractions that have unlike denominators are a
different matter. You have to find something called the
common denominator before you can add or subtract.
The bottom numbers must be the same to add or
subtract fractions.
You may remember having to find the least common
denominator in school. I know some of you hated that
lesson… and still do! So, we are going to use a shortcut.
Don’t you tell your former math teachers!
When you have to add or subtract fractions, just
multiply the 2 bottom numbers of the fractions
together. The result is a common denominator but not
necessarily the “least common denominator.” This is
what we will do when we add and subtract fractions.
If you need a denominator to add
1 3
+ , you will
3 20
use 60 (3 × 20 = 60).
Applied Mathematics • 19
LESSON 2
For
3 2
− , you can use 42 (7 × 6 = 42).
7 6
For
5
1
+ , you can use 240 (12 × 20 = 240).
12 20
240 is not the smallest number we could use. So, if
you know how to find the least common denominator,
please do so. (The LCD for this problem is 60). But, if
you have forgotten how to find it, just use our shortcut
and multiply the 2 bottom numbers of the fractions.
Using the shortcut, we will have larger numbers that
need to be reduced, but we can use our calculators to
help us reduce.
Speaking of calculators… if you have a fraction key,
, you do not need to worry about denominators
or reducing fractions. If you struggle with or dislike
fractions, I would suggest you think about purchasing
a calculator with this key. (They are rather inexpensive
and may be found in electronics at major discount
stores.)
20 • Applied Mathematics
LESSON 2
Let’s add a fraction with unlike denominators:
1 5
+
8 6
For those of you with calculators that work fractions, enter:
and your display should indicate 23 24 which is
23
.
24
If you do not have this key, multiply the 2 bottom numbers together:
8 × 6 = 48
We will use 48 as our denominator. Now, we must rewrite our
fractions with this new denominator. To rewrite the fractions, set it
up like:
1 ?
=
8 48
5 ?
=
6 48
1
into some fraction with 48 on the bottom.
8
Divide 8 into 48. It goes 6 times. Now, multiply 6 by the 1 (top
number) to get 6.
We want to change
1 6
=
8 48
This means
1
6
is the same measurement as
.
8
48
Applied Mathematics • 21
LESSON 2
Now, let’s change
5
.
6
5 ?
=
6 48
Divide 6 into 48 to get 8. Multiply 8 by 5 to get 40.
5 40
=
6 48
Since both fractions are now written with “like” denominators, we
can add.
6 40 46
+
=
48 48 48
Reduce answer (divide both top and bottom by the same number…
if the number goes evenly when you divide, it will reduce).
46
48
If both numbers are even, 2 will always divide into both:
46 ÷ 2 = 23
48 ÷ 2 = 24
So,
46 23
=
48 24
22 • Applied Mathematics
LESSON 2
Sometimes you have to reduce the number again.
This fraction will not reduce again. As you may remember, this is the
same answer as we found earlier using a calculator.
If you know how to find the least common denominator, you would
have used 24 instead of 48.
1 ?
=
8 24
5 ?
=
6 24
Rewrite fractions:
1 3
=
8 24
5 20
=
6 24
So,
3 20 23
+
=
24 24 24
In this case by using the lowest common denominator, you saved
yourself from having to reduce. (But, always check your answer to
see if you can reduce.)
Ready to try another problem?
Applied Mathematics • 23
LESSON 2
By using a fraction key:
4 3
−
5 15
Key in:
4 3 3
− =
5 15 5
By multiplying denominators:
5 × 15 = 75
4 ?
=
5 75
3
?
=
15 75
5 goes into 75, 15 times, 15 × 4 = 60
15 goes into 75, 5 times, 5 × 3 = 15
Tip: The number of times
one goes into the other is
always the opposite
denominator when using
our shortcut.
24 • Applied Mathematics
LESSON 2
4 60
=
5 75
3 15
=
15 75
60 15 45
−
=
75 75 75
To reduce, you might divide by 5.
If both numbers end with a 5 or a 0, then 5 will divide into both.
45 ÷ 5 9
=
75 ÷ 5 15
This is not the final answer! It will reduce again, this time by 3:
9÷3 3
=
15 ÷ 3 5
3
is the final answer.
5
We could have divided our original answer,
45
, by 15:
75
45 ÷ 15 3
=
75 ÷ 15 5
Applied Mathematics • 25
LESSON 2
It does not matter which way you reduce. It does
matter that you do not make careless mistakes and that
you always reduce as far as possible (until no more
common numbers divide evenly into each one).
By not finding the lowest or least common
denominator, we are causing ourselves some extra work
in reducing. We are all different and you have to decide
which way is easiest for you.
By finding the least common denominator:
4 ?
=
5 15
4 12
=
5 15
3
?
=
15 15
3
3
=
15 15
Anytime one bottom number divides evenly into the other (15 ÷ 5 =
3), you can use the larger number as the common denominator.
12 3
9
− =
15 15 15
Both top and bottom numbers divide by 3.
9÷3=3
15 ÷ 3 = 5
So,
9 3
=
15 5
26 • Applied Mathematics
LESSON 2
You may also have some fractions with a whole
number in the front. This is called a mixed number.
(Part whole number, part fraction.) One way to work
with mixed numbers is to change this number to
something called an improper fraction.
Example:
4
5
changed to improper fraction:
6
Multiply the whole number and the denominator and then add the top the numerator.
We would convert to an improper fraction to add the following fractions.
5 29
4 =
6 6
1 7
+3 =
2 2
(3 × 2 = 6 → 6 + 1 = 7)
This way, you won’t have to keep track of whole
numbers and fractions. There are other ways to add
mixed numbers. If you prefer another method, do not
try to learn a new method, just check your final answers
with mine.
Applied Mathematics • 27
LESSON 2
After converting from mixed numbers to improper
fractions, you must find like denominators. 2 × 6 = 12
or, since 2 divides evenly into 6 we could use 6 as the
common denominator.
29 29
=
6
6
7 21
+ =
2 6
29 21 50
+ =
6
6
6
50 25
= (reduceby 2)
6
3
To change the answer back to a mixed number,
divide the bottom into the top number:
25 ÷ 3
Your calculator will show a decimal answer 8.3333…
Since we are working in fractions, you will need to divide this problem
by hand.
8
3 25
)
−24
1
8 is the whole number, 1 is the remainder.
So, you have 8
28 • Applied Mathematics
1
as an answer.
3
LESSON 2
To enter the addition problem with mixed numbers, press:
The display should be similar to:
8 1
1
3 which means 8 . Converted… added… and reduced.
3
I like that
key!
Of course the easiest way to add mixed numbers is to let your
calculator do the work!
If you have a problem with a whole number by itself,
you need to remember that every whole number has a
1 under it. For instance:
4=
4
1
16 =
16
1
Applied Mathematics • 29
LESSON 2
For example:
If you need to subtract
5−
2
from 5:
3
2
3
You can rewrite 5 as
5
.
1
5
1
2
−
3
You need a common denominator. If one of the bottom
numbers is 1, your common denominator will always
be the other number. In this case, 3.
5 ?
=
1 3
2 ?
=
3 3
30 • Applied Mathematics
LESSON 2
Convert to the new top number:
5 15
=
1 3
2 2
= (stays same since it already had 3 in denominator)
3 3
15 2 13
− =
3 3 3
4
3 13
)
12
1
1
The answer is 4 .
3
You should probably practice some on your own now.
I’ll throw in some word problems, too. Remember to
check your answers.
Applied Mathematics • 31
LESSON 2
EXERCISE – ADDING AND SUBTRACTING FRACTIONS
Instructions:
Add or subtract the following fractions. Reduce fractions to lowest terms and
write final answers as mixed numbers.
1.
4 3
+ =
5 5
2.
1 5
+ =
6 6
3.
4 2
− =
7 7
4.
3 4
− =
2 5
5.
6
1
4 +2 =
7
3
32 • Applied Mathematics
LESSON 2
6.
Before working any more problems, write a step-by-step procedure
explaining how to add and subtract fractions which have unlike
denominators. (Refer to the instructions if you have trouble.)
7.
5−3
8.
7 5
− =
3 7
9.
5
7
=
9
6
4
+2
=
11
33
Applied Mathematics • 33
LESSON 2
10.
A tank holds 800 gallons of water. The gauge indicates that the
3
4
tank is
pool
7
8
full; then 100 gallons are drained from the tank to fill a
full. Next,
1
3
of a tank of water is added back in. What fraction
of the tank is filled with water?
11.
A stick is 4
1
2
ft long. A 3 ft portion of the stick is decayed. How
much of the stick is not decayed?
12.
A grandmother left her estate to her four grandchildren. She left
to Jill,
1
3
to Jack,
1
6
to Hansel and the remainder was left to Gretel.
What fraction of the estate did Gretel receive?
34 • Applied Mathematics
1
2
LESSON 2
ANSWERS TO EXERCISE
1.
4 3
+ =
5 5
Answer:
4 +3
5
7
2
=1
5
5
2.
1 5
+ =
6 6
Answer:
1+5
6
6
=1
6
3.
4 2
− =
7 7
Answer:
4−2
7
2
7
Applied Mathematics • 35
LESSON 2
4.
3 4
− =
2 5
Answer:
convert to fractions with common denominator:
3 15
4 8
=
=
and
2 10
5 10
subtract:
15 8
15 − 8
−
→
10 10
10
7
10
36 • Applied Mathematics
LESSON 2
5.
6
1
4 +2 =
7
3
Answer:
convert to improper fractions:
6 34
1 7
4 =
and 2 =
7 7
3 3
find common denominator:
34 ?
7 ?
=
=
and
7
21
3 21
find equivalent fractions:
34 102
7 49
=
=
and
3 21
7
21
add top numbers:
102 49 102 + 49
+
=
21 21
21
151
4
=7
21
21
6.
Before working any more problems, write a step-by-step procedure
explaining how to add and subtract fractions which have unlike
denominators. (Refer to the instructions if you have trouble.)
Answer:
Your answer will be in your own words, so it may be different
from someone else’s. Look back at the instructions to check
your answer.
Applied Mathematics • 37
LESSON 2
7.
5−3
7
=
9
5=
Answer:
−3
8.
7 34 34
=
=
9 9
9
11
2
=1
9
9
7 5
− =
3 7
Answer:
9.
5 45
=
1
9
5
7 49
=
3 21
5 15
− =
7 21
34
13
=1
21
21
6
4
+2
=
11
33
Answer:
38 • Applied Mathematics
6 61 183
=
=
11 11 33
4 70 70
+2
=
=
33 33 33
253
2
=7
33
3
5
LESSON 2
10.
A tank holds 800 gallons of water. The gauge indicates that the
tank is
pool
7
8
3
4
full; then 100 gallons are drained from the tank to fill a
full. Next,
1
3
of a tank of water is added back in. What fraction
of the tank is filled with water?
Answer:
Tank is
3
4
full. Subtract fraction of water drained… given in
gallons.
100(drained) 1
=
800(of total) 8
7
8
Ignore …this is referring to a pool… you are interested in
what is in the tank. Then add
1
3
to the remainder of the
subtraction problem.
(continued)
Thinking about word
problems
Applied Mathematics • 39
LESSON 2
Problem:
1
1
3
(starting capacity) − (100 gallons) + (added to tank)
4
8
3
Rewrite fractionswith like denominators:
3 72
1 12
1 32
=
→
=
→
=
4 96
8 96
3 96
We now have:
72 12 32 92
−
+
=
12 96 96 96
Reduce:
92 46 23
=
=
96 48 24
The tank is
40 • Applied Mathematics
23
full.
24
LESSON 2
11.
A stick is 4
1
2
ft long. A 3 ft portion of the stick is decayed. How
much of the stick is not decayed?
1 9 9
= =
2 2 2
3 6
−3 = =
1 2
3
1
= 1 ft
2
2
4
Answer:
or
4
1
2
−3
1
1
ft
2
Applied Mathematics • 41
LESSON 2
12.
A grandmother left her estate to her four grandchildren. She left
to Jill,
1
3
to Jack,
1
6
1
2
to Hansel and the remainder was left to Gretel.
What fraction of the estate did Gretel receive?
Answer:
1 1 1
+ +
2 3 6
Rewrite fractions:
1 3
=
2 6
1 2
=
3 6
1 1
=
6 6
add like fractions:
3 2 1 6
+ + = =1
6 6 6 6
1, a whole number, means the whole estate. So, there
was nothing left to give to Gretel.
42 • Applied Mathematics
LESSON 3
INTRODUCTION TO NEGATIVE NUMBERS
Lesson 3 will give you an overview of positive and
negative numbers. You’ve always dealt with positive
numbers, but you may think you have never dealt with
negative numbers. Think about it for a minute, though.
You’ve actually always dealt with negative numbers, too.
Have you ever withdrawn money from a bank? That’s
a negative sum. Maybe you haven’t thought about how
to add or subtract them.
We often refer to positive and negative numbers as
signed numbers. A number line is helpful in working
with signed numbers.
This is a number line:
The numbers on the right are positive, and the
numbers on the left are negative. Zero does not have a
sign; it is neutral. Notice that the positive numbers do
not have a sign. A positive number can be written with
or without a sign. (for example, 5 or +5). The farther
to the right you go, the bigger the number. The farther
to the left you go, the smaller the number.
If you think about it, you already know how to add
positive numbers.
4+5=9
Applied Mathematics • 43
LESSON 3
Sometimes, though, the signs get a little confusing.
For example, you might have a problem that looks like
this:
+4 + (+5) =
Now, that’s exactly what we have done already but
it looks a little strange. When you see a problem like
that, concentrate on finding two signs that are together:
Identify:
Once you have located 2 signs together, check to
see if the signs are the same or different.
44 • Applied Mathematics
LESSON 3
If they are the same, change the sign to a “plus.” If
they are different, change the sign to a “minus.” Now
your problems look like this:
Notice that the signs in the middle have all been
changed to reflect the previous rule.
We still haven’t added or subtracted yet. Place the
numbers “up and down.”
Applied Mathematics • 45
LESSON 3
Now, if the signs of both numbers are the same,
you should add the numbers and carry the sign down:
If the signs are different, subtract and use the sign
of the larger number:
46 • Applied Mathematics
LESSON 3
It sounds a little complicated, but it just takes practice.
These rules are for anyone who does not have a
calculator. I hope you do, then signed numbers will be
much easier to use. Your calculator will add and subtract
signed numbers, but you must know how to enter the
information.
So, let’s look at our calculators. You should have a
button or key that looks like this if your calculator will
handle signed numbers:
Applied Mathematics • 47
LESSON 3
or
is not the same as the
have the
or
key. If you do not
key, you will have to use the
signed number rules or invest in a calculator that
computes signed numbers.
Now, try some on your own. Remember the answers
are provided, but don’t peek!
Positive and negative
numbers relate to earning
and spending money.
48 • Applied Mathematics
LESSON 3
EXERCISE – ADDING NEGATIVE AND POSITIVE NUMBERS
Instructions:
Use the negative key on the calculator to solve the following problems. If you
do not have a calculator with this key, use the rules for signed numbers.
1.
-5 + 2 =
2.
4 + (-4) =
3.
-0.8 + (-0.6) =
4.
-5 + (-7) =
5.
-7 + (-4) =
6.
-46 + 86 =
7.
-471 + (-399) =
8.
0 + (-4.3) =
Applied Mathematics • 49
LESSON 3
9.
4 + (-3.5) =
10.
−
4 3
+ =
3 7
5
8
2
+ −8
3
−4
11.
−45
12.
+
5
6
7
8
50 • Applied Mathematics
LESSON 3
ANSWERS TO EXERCISE
1.
-5 + 2 =
Answer:
2.
-12
same sign, so add
keep the sign
-11
same sign, so add
keep the sign
40
opposite signs, so subtract
keep sign of larger number
-46 + 86 =
Answer:
7.
-1.4 same sign, so add
keep the sign
-7 + (-4) =
Answer:
6.
opposite signs, so subtract
-5 + (-7) =
Answer:
5.
0
-0.8 + (-0.6) =
Answer:
4.
opposite signs, so subtract
keep sign of larger number
4 + (-4) =
Answer:
3.
-3
-471 + (-399) =
Answer:
-870 same sign, so add
keep the sign
Applied Mathematics • 51
LESSON 3
8.
0 + (-4.3) =
Answer:
9.
4 + (-3.5) =
Answer:
10.
-4.3 different signs, so subtract
keep sign of larger number
−
.5
opposite signs, so subtract
keep sign of larger number
4 3
+ =
3 7
Answer:
−
28 9
19
+
=−
21 21
21
or
-19 21
52 • Applied Mathematics
LESSON 3
5
8
2
+ −8
3
−4
11.
Answer:
same sign, so add
Change to improper fractions:
5
37
=−
8
8
2
26
+ −8 = −
3
3
−4
Find common denominator and change fractions:
37
?
= −
8
24
26
?
−
= −
3
24
−
add top numbers
Keep the sign:
37
111
=−
8
24
26
208
−
=−
3
24
319
−
24
−
Convert to mixed number:
−13
7
or…(#11. continued on next page)
24
Applied Mathematics • 53
LESSON 3
11.
(continued)
Answer:
…or
-13
7 24
which is −13
54 • Applied Mathematics
7
24
LESSON 3
−45
12.
+
5
6
7
8
Answer:
opposite signs so subtract
Change to improper fractions (first number):
5
275
=−
6
6
7
7
+ =
8
8
−45
Find common denominator and change fractions:
275
−2, 200
=
6
48
7
42
+
=
8
48
−2,158
48
−
which changed to a mixed number
and reduced is −44
46
23
= −44
48
24
or use your calculator to find:
-44 23 24
Applied Mathematics • 55
LESSON 3
EXERCISE – SUBTRACTING POSITIVE AND NEGATIVE NUMBERS
Instructions:
Use the negative key on the calculator to solve the following problems. If you
do not have a calculator with a negative key, use the rules for signed numbers.
1.
-4 - 5 =
2.
5-6=
3.
5 - (-6) =
4.
13 - (-13) =
5.
58 - 13 =
6.
-12 - 15 =
56 • Applied Mathematics
LESSON 3
7.
-0.99 - 1 =
8.
-56 - 45 =
9.
-88 - (-55) =
10.
0.48 - 2 =
11.
-4.6 - 7.6 =
12.
-4.07 - (-2.01) =
13.
3.5 - (-3) =
Applied Mathematics • 57
LESSON 3
14.
10 - (-3) =
15.
2
9
−5 − 8
=
7
10
16.
−
17.
12 −
18.
1 12
2 −
=
2 7
3 8
− =
5 9
5
=
6
58 • Applied Mathematics
LESSON 3
ANSWERS TO EXERCISE
1.
-4 - 5 =
Answer:
2.
5-6=
Answer:
3.
-1.99
-56 - 45 =
Answer:
9.
-27
-0.99 - 1 =
Answer:
8.
45
-12 - 15 =
Answer:
7.
26
58 - 13 =
Answer:
6.
11
13 - (-13) =
Answer:
5.
-1
5 - (-6) =
Answer:
4.
-9
-101
-88 - (-55) =
Answer:
-33
Applied Mathematics • 59
LESSON 3
10.
0.48 - 2 =
Answer:
11.
-4.6 - 7.6 =
Answer:
12.
13
2
9
−5 − 8
=
7
10
Answer:
16.
6.5
10 - (-3) =
Answer:
15.
-2.06
3.5 - (-3) =
Answer:
14.
-12.2
-4.07 - (-2.01) =
Answer:
13.
-1.52
−
−14
13
70
3 8
− =
5 9
Answer:
60 • Applied Mathematics
−1
22
45
LESSON 3
17.
12 −
5
=
6
Answer:
18.
11
1
6
1 12
2 −
=
2 7
Answer:
11
14
Applied Mathematics • 61
LESSON 3
EXERCISE – PROBLEMS INVOLVING SIGNED NUMBERS
Instructions:
Solve the following word problems.
1.
The temperature in Atlanta, Georgia is 78 degrees. The temperature
in Omaha, Nebraska is -12 degrees. What is the temperature
difference in the two cities?
2.
Death Valley is 280 feet below sea level. Lookout Mountain is 2,391
feet above sea level. What is the difference in their altitudes?
3.
A plane is flying at an altitude of 36,000 feet. The pilot notices
turbulent clouds are ahead, so he ascends the plane another 4,000
feet. Later, he lowers the plane by 5,500 feet. What is the new altitude
of the plane?
62 • Applied Mathematics
LESSON 3
4.
Come On In gift shop owes money totaling $2,345.61 to its
employees, the bank, the wholesaler, and the plumber. One of the
gift shop’s best customers, Sammy Sorry, owes the gift shop
$560.55. Ignoring any other assets or charge customers, what is
the gift shop’s net assets?
5.
Sheri wrote the following entries in her checkbook: -$13.52, -$15.88,
+$500, -$451.57, +$275, -$244.58, + $2,516. What is the total of her
deposits? What is the average amount of her checks? What overall
change in her balance would occur as a result of these 7
transactions?
Applied Mathematics • 63
LESSON 3
ANSWERS TO EXERCISE
1.
The temperature in Atlanta, Georgia is 78 degrees. The temperature
in Omaha, Nebraska is -12 degrees. What is the temperature
difference in the two cities?
Answer:
Key word - difference - subtract
your display should show:
90 or
78 + 12 = 90˚
2.
Death Valley is 280 feet below sea level. Lookout Mountain is 2,391
feet above sea level. What is the difference in their altitudes?
Answer:
Key word - difference - subtract
2,391 + 280 =
2,671 ft
3.
A plane is flying at an altitude of 36,000 feet. The pilot notices
turbulent clouds are ahead, so he ascends the plane another 4,000
feet. Later, he lowers the plane by 5,500 feet. What is the new altitude
of the plane?
Answer:
64 • Applied Mathematics
36,000 + 4,000 - 5,500 =
40,000 - 5,500 =
34,500 ft
LESSON 3
4.
Come On In gift shop owes money totaling $2,345.61 to its
employees, the bank, the wholesaler, and the plumber. One of the
gift shop’s best customers, Sammy Sorry, owes the gift shop
$560.55. Ignoring any other assets or charge customers, what is
the gift shop’s net assets?
Answer:
5.
-$2,345.61 + $560.55 = -$1,785.06
Sheri wrote the following entries in her checkbook: -$13.52, -$15.88,
+$500, -$451.57, +$275, -$244.58, + $2,516. What is the total of her
deposits? What is the average amount of her checks? What overall
change in her balance would occur as a result of these 7
transactions?
Answer:
Deposits (+)
$500 + $275 + $2,516 = $3,291
Check average:
$13.52 + $15.88 + $451.57 + $244.58 = $725.55
Total of $725.55 for 4 checks
Find average amount of checks:
$725.55 ÷ 4 = $181.39
Find overall change:
$3,291 (deposits) - $725.55 (checks) = $2,565.45
Her balance increased $2,565.45
Applied Mathematics • 65
LESSON 4
MULTIPLYING AND DIVIDING WITH
NEGATIVE NUMBERS
You’ve added and subtracted positive and negative
numbers. Now, it’s time to multiply and divide. (This
is easy even if you do not have a calculator!)
Remember, in addition and subtraction how I told
you to look for two signs? I said that if they were alike,
they would be replaced with a positive; and if they’re
different, replace with a negative. Those are the only
rules you have to remember. (Same signs = +, different
signs = -).
-5 × -4 = +20
(If the signs are the same, the answer will be a “+.” If
they’re different, the answer will be a “-”).
-3 × 2 = -6
9 ÷ -3 = -3
-9 ÷ -3 = 3
See how easy this is! Now, you try some.
Hope you speed through
this lesson.
66 • Applied Mathematics
LESSON 4
EXERCISE – MULTIPLYING AND DIVIDING SIGNED NUMBERS
Instructions:
Use the negative key on the calculator to solve the following problems. If you
do not have a calculator with a negative key, use the rules for signed numbers.
1.
4×3=
2.
4 × (-3) =
3.
-4 × 3 =
4.
3.4 × (-4.5) =
5.
-0.9 × (-4) =
6.
-1 × (-1) × (-1) =
7.
4 × (-1) × (-3) =
8.
-3 × (-2) =
Applied Mathematics • 67
LESSON 4
9.
8 × (-2) =
10.
4  −3 
×  =
5  8
11.
 −3   7 
  ×  =
 4   4
12.
 1  3 
 −3  ×  5  =
 3  4
13.
-4 ÷ -2 =
14.
-45 ÷ 5 =
15.
-8 ÷ -0.5 =
68 • Applied Mathematics
LESSON 4
16.
-45.81 ÷ 0.3 =
17.
-0.81 ÷ -9 =
18.
68 ÷ 4 =
Applied Mathematics • 69
LESSON 4
ANSWERS TO EXERCISE
1.
4×3=
Answer:
2.
4 × (-3) =
Answer:
3.
-15.3
-0.9 × (-4) =
Answer:
6.
-12
3.4 × (-4.5) =
Answer:
5.
-12
-4 × 3 =
Answer:
4.
12
3.6
-1 × (-1) × (-1) =
Answer:
-1
-1 × -1 = +1
1 (result of first 2 numbers) × -1 = -1
7.
4 × (-1) × (-3) =
Answer:
8.
12
-3 × (-2) =
Answer:
70 • Applied Mathematics
6
LESSON 4
9.
8 × (-2) =
Answer:
10.
4  −3 
×  =
5  8
Answer:
11.
5
−1
16
−19
1
6
-4 ÷ -2 =
Answer:
14.
3
10
 1  3 
 −3  ×  5  =
 3  4
Answer:
13.
−
 −3   7 
  ×  =
 4   4
Answer:
12.
-16
2
-45 ÷ 5 =
Answer:
-9
Applied Mathematics • 71
LESSON 4
15.
-8 ÷ -0.5 =
Answer:
16.
-45.81 ÷ 0.3 =
Answer:
17.
-152.7
-0.81 ÷ -9 =
Answer:
18.
16
.09
68 ÷ 4 =
Answer:
72 • Applied Mathematics
17
LESSON 4
EXERCISE – MIXED OPERATIONS WITH SIGNED NUMBERS
Instructions:
Use the negative key on the calculator to solve the following problems. If you
do not have a calculator with a negative key, use the rules for signed numbers.
1.
-4 + (-5.8) =
2.
−3 −4
−
=
4
5
3.
2-5=
4.
8 × (-3) =
5.
-591 ÷ -3 =
6.
81 - (-45) =
Applied Mathematics • 73
LESSON 4
7.
4 3
× =
7 4
8.
5.8 ÷ (-2.9) =
9.
-0.5 × 4.2 =
74 • Applied Mathematics
LESSON 4
ANSWERS TO EXERCISE
1.
-4 + (-5.8) =
Answer:
2.
−3 −4
−
=
4
5
Answer:
3.
-24
-591 ÷ -3 =
Answer:
6.
-3
8 × (-3) =
Answer:
5.
1
20
2-5=
Answer:
4.
-9.8
197
81 - (-45) =
Answer:
126
Applied Mathematics • 75
LESSON 4
7.
4 3
× =
7 4
Answer:
8.
5.8 ÷ (-2.9) =
Answer:
9.
12 3
=
28 7
-2
-0.5 × 4.2 =
Answer:
76 • Applied Mathematics
-2.1
LESSON 4
Now, when will you ever need to use signed
numbers? We are going to apply what we have learned
as soon as we review the steps for problem solving. After
that, I will let you practice some word problems with
signed numbers.
PROBLEM SOLVING STRATEGIES AND TECHNIQUES
1. Analyze the situation to understand the problem.
In a situation where the problem is a written problem, read the problem
thoroughly and determine what the problem is asking you to find. Actual work
related problems need to be thought out thoroughly as well.
2. Determine what information you must know to solve the problem. If the
problem does not provide the information, determine how you can find the
information.
3. Determine what must be done with that information and do it.
4. Check your answer to be certain it is a reasonable answer and to be certain that
it is the solution to the original problem.
TECHNIQUES
Read the problem twice. On the first reading, be sure you understand the problem
and know what it is asking you to find. On the second reading, pick out the
information you will use to solve the problem.
Anytime you are solving a problem with any type of shape draw a rough sketch
and label the information you are given on your drawing.
Round the numbers in the problem to “nice” numbers to get a rough estimate of
what the answer should be to determine if your answer is “in the ballpark.”
Use a variable to represent the piece(s) of information which you do not know.
Applied Mathematics • 77
LESSON 4
EXERCISE – APPLICATIONS INVOLVING MIXED OPERATIONS WITH
SIGNED NUMBERS
Instructions:
Solve the following word problems.
1.
An airplane is descending at a rate of 5,000 ft/min for 3 minutes.
How much higher was the plane before it started descending?
2.
A tank is leaking gasoline at a rate of 4 gal/day. How long will it take
to lose 50 gallons of gasoline?
3.
The payroll department underpaid fifteen employees from
department 18 by $17.12 each on January 31. What is the total error
and is it a deficit or credit for the payroll department?
78 • Applied Mathematics
LESSON 4
4.
The IRS claims that 250 people in the 37421 zip code area have
filed returns that average $1,245 in underpayment. The paycheck
for each of the people, who work at 52 different companies in the
Chattanooga area, must be adjusted to avoid a penalty for
underpayment in the following year. What is the total amount of
adjustment in these paychecks?
5.
The temperatures for five consecutive days in a month with 31 days
are -8 degrees, -15 degrees, -5 degrees, 13 degrees, and 5 degrees.
What was the average temperature for these days?
6.
The bank charged Derek a $15.00 service charge for each of 7
checks returned for insufficient funds over a period of 13 days.
What is the total service charge and how will he enter this amount
to balance his checkbook?
Applied Mathematics • 79
LESSON 4
ANSWERS TO EXERCISE
1.
An airplane is descending at a rate of 5,000 ft/min for 3 minutes.
How much higher was the plane before it started descending?
Answer:
You may use a proportion to determine how many feet the
airplane dropped.
5,000 N
=
1
3
5,000 × 3 = 1 × N
15,000 = N
The question was how many feet higher was the airplane
before the descent. So, the answer is a positive 15,000 or
15,000 feet higher.
2.
A tank is leaking gasoline at a rate of 4 gal/day. How long will it take
to lose 50 gallons of gasoline?
Answer:
4 gal
1day
=
50 gal
N
4N = 50
N = 12.5
The tank will lose 50 gallons of gasoline in 12.5 days.
80 • Applied Mathematics
LESSON 4
3.
The payroll department underpaid fifteen employees from
department 18 by $17.12 each on January 31. What is the total error
and is it a deficit or credit for the payroll department?
Answer:
15 employees underpaid by $17.12 each
Department 18 and date are extraneous information (do not
need it to solve problem)
Key words - total (of equal amounts) so we multiply
15 × 17.12 = 256.8
The total error was for $256.80. This money is withdrawn, so
it is a deficit for the payroll department or -$256.80.
4.
The IRS claims that 250 people in the 37421 zip code area have
filed returns that average $1,245 in underpayment. The paycheck
for each of the people, who work at 52 different companies in the
Chattanooga area, must be adjusted to avoid a penalty for
underpayment in the following year. What is the total amount of
adjustment in these paychecks?
Answer:
250 people have underpayments that average $1,245
Zip code & 52 companies are extraneous information (do
not need it to solve problem)
Key words - total (of equal amounts) so we multiply
250 × 1,245
$311,250 needs to be made in adjustments
Applied Mathematics • 81
LESSON 4
5.
The temperatures for five consecutive days in a month with 31 days
are -8 degrees, -15 degrees, -5 degrees, 13 degrees, and 5 degrees.
What was the average temperature for these days?
Answer:
Find an average, so add temperatures and divide by the 5
days recorded
31 days is extraneous information (do not need it to solve
problem)
-8 + -15 + -5 + 13 + 5 = -10
-10 (total) / 5 (days) = -2 degrees average temperature
6.
The bank charged Derek a $15.00 service charge for each of 7
checks returned for insufficient funds over a period of 13 days.
What is the total service charge and how will he enter this amount
to balance his checkbook?
Answer:
$15.00 service charge for 7 checks
13 day period is extraneous information (do not need it to
solve problem)
15 × 7 = 105
Derek paid $105 service charge. He should subtract (-$105)
this amount from his account.
82 • Applied Mathematics
LESSON 5
REVIEW OF PERCENT PROBLEMS
Lesson 5 should be a review for you. In this lesson,
we will briefly review percentages. Remember that the
word “of ” means multiply, and the word “is” means
equal. When we work percentage problems, we will
write equations by using the following symbols:
what →
is
→
of
→
N
=
×
Remember you must change percents to decimals before
you multiply or divide these problems. (Use the percent
key on your calculator if you have one.)
Example 1:
What is 140% of 45?
N = 140% × 45
(140% and 45 are on same side of = sign, so multiply)
N = 63
Example 2:
15 is what percent of 25?
15 = N% × 25
(divide by the number on the same side of = sign as N)
15
=N
25
Since the question asked for a percent, the decimal must be moved
two places to the right.
0.60 = 60% is your answer.
You’ve already done problems like this. Now, you should
review by practicing.
Applied Mathematics • 83
LESSON 5
Pop Quiz: Your supervisor asks you to have a 36 exposure roll of film developed.
He knows that he will need double prints on 9 of the pictures. The film
development company you use gives your company a discount for double prints. If
you have double prints made of the whole roll, they offer a 15% discount. The
regular price for double prints is $0.45 per exposure and the regular price for
single prints is $0.32 per exposure. You must decide which would be the most
economical way to get the film developed - double prints of the whole roll to
begin with or single prints at first then get 9 reprints at $0.51 each later.
What will be the cost using the least expensive method?
84 • Applied Mathematics
LESSON 5
EXERCISE – SOME GENERIC PERCENT PROBLEMS FOR REVIEW
Instructions:
Solve the following percent problems. Round numbers to the nearest tenth.
1.
What is 55% of 120?
2.
What is 40% of 35?
3.
84 is what percent of 96?
4.
What is 45% of 900?
5.
15% of what is 24?
6.
16 is what percent of 80?
7.
What percent of 50 is 65?
Applied Mathematics • 85
LESSON 5
ANSWERS TO EXERCISE
1.
What is 55% of 120?
Answer:
2.
What is 40% of 35?
Answer:
3.
N = 55% × 120
N = 66
N = 40% × 35
N = 14
84 is what percent of 96?
Answer:
84 = N% × 96
84
= N%
96
= N%
87.5% = N%
4.
What is 45% of 900?
Answer:
5.
N = 45% × 900
N = 405
15% of what is 24?
Answer:
15% × N = 24
.15 × N = 24
24
.15
N = 160
N=
86 • Applied Mathematics
LESSON 5
6.
16 is what percent of 80?
Answer:
16 = N% × 80
16
= N%
80
= N%
20% = N%
7.
What percent of 50 is 65?
Answer:
N% × 50 = 65
N% =
65
50
N% =
N% = 130%
Applied Mathematics • 87
LESSON 5
Remember when you do word problems involving
percents, you should try to fill in the missing part of
this sentence.
“What percent of the total is some number?”
Here’s an example:
A shirt was originally marked $40.00, but was marked down to
$32.00. What is the percent of the discount?
We don’t know the percent, but we know the original total. We also
know that the shirt was marked down $8. (40 - 32 = $8) Using:
What % of the total is some number?
we can fill in:
What % of $40 is $8?
N% × $40 = $8
N% = 8/40
N% = .2
N% = 20%
Try the problems in the next exercise like this. The
hard part will be deciding what will go in the blanks.
88 • Applied Mathematics
LESSON 5
EXERCISE – PERCENT APPLICATIONS FOR REVIEW
Instructions:
Use the sentence “What percent of the total is some number?” to solve the
percent problems. Round final decimal answers to the nearest tenth.
1.
If the construction of an office building is budgeted at $150,000
and 6% of that is allowed for painting expenses, how much can be
spent on painting?
2.
A solution must be 14% insecticide. If you must mix 5 gallons of
this solution, how much insecticide must you use?
3.
There are 40 support personnel at a company. If 35 of them have a
bachelor’s degree, what percent of them have a bachelor’s degree?
4.
The goal this month is to have a 4% increase in production over
last month. If there were 450 products produced last month and
466 the month before that, how many products must be produced
this month to achieve this goal?
Applied Mathematics • 89
LESSON 5
5.
A customer of yours receives a discount on each purchase. If her
bill was $1,600 before the discount was applied, and it was $1,460
after the discount was applied, what percent discount does she
receive?
Customers like for me to
“cut” prices.
90 • Applied Mathematics
LESSON 5
ANSWERS TO EXERCISE
1.
If the construction of an office building is budgeted at $150,000
and 6% of that is allowed for painting expenses, how much can be
spent on painting?
Answer:
2.
A solution must be 14% insecticide. If you must mix 5 gallons of
this solution, how much insecticide must you use?
Answer:
3.
6% × 150,000 = N
$9,000 = N
14% × 5 = N
.7 gallons = N
There are 40 support personnel at a company. If 35 of them have a
bachelor’s degree, what percent of them have a bachelor’s degree?
Answer:
N% × 40 = 35
N% =
N% =
4.
35
40
= 87.5%
The goal this month is to have a 4% increase in production over
last month. If there were 450 products produced last month and
466 the month before that, how many products must be produced
this month to achieve this goal?
Answer:
4% × 450 = N
(466 is not relevant to the question about last month)
18 = N (this is how many more products need to be produced
for an increase of 4%)
18 + 450 (last month) = 468 products to achieve goal
Applied Mathematics • 91
LESSON 5
5.
A customer of yours receives a discount on each purchase. If her
bill was $1,600 before the discount was applied, and it was $1,460
after the discount was applied, what percent discount does she
receive?
Answer:
1,460 is some number but it is not the amount we need to
solve this problem. This is how much the customer paid.
The discount amount is:
$1,600 - $1,460 = $140
N% × 1,600 = 140
140 (decrease)
N% =
1,600 (original amount)
N% =
N% = 8.75% discount rounds to 8.8%
92 • Applied Mathematics
LESSON 5
Hey partner! You might want to
memorize these!
Applied Mathematics • 93
LESSON 6
SOLVING MULTIPLE RATE PROBLEMS
Lesson 6 deals specifically with rate problems. The
first set should be a review for you, but let me do one
example to refresh your memory.
You must make a 556 mile road trip. If you average 63 mph, how long will
it take you to reach your destination?
63 miles
1 hour
=
556 miles
N hours
Then cross multiply:
63N = 556
556
63
N = 8.8 hours
N=
It will take almost 9 hours at this speed.
You should review calculating rates by working the
following problems.
94 • Applied Mathematics
LESSON 6
EXERCISE – REVIEW OF RATE PROBLEMS
Instructions:
Use proportions to solve the following problems. Round answers to the nearest
tenth.
1.
A car goes 50 mph for 4.4 hours. How far does it travel?
2.
A worker can produce 5.4 widgets per hour. If she works 40 hours
per week and gets two weeks of vacation each year, how many
widgets would you expect her to produce in a year? (Assume no
holidays.)
3.
The office manager can type 89 words per minute. How long would
it take her to type a document containing 1,045 words?
Applied Mathematics • 95
LESSON 6
ANSWERS TO EXERCISE
1.
A car goes 50 mph for 4.4 hours. How far does it travel?
Answer:
50
N  miles 
=


1
4.4  hours 
N = 220 miles
2.
A worker can produce 5.4 widgets per hour. If she works 40 hours
per week and gets two weeks of vacation each year, how many
widgets would you expect her to produce in a year? (Assume no
holidays.)
Answer:
40 hours × 50 weeks = 2,000 hours
 widgets 
5.4
N
=


1
2,000  hours 
N = 10,800 widgets
3.
The office manager can type 89 words per minute. How long would
it take her to type a document containing 1,045 words?
Answer:
96 • Applied Mathematics
89 1,045  words 
=


 minutes 
1
N
89N = 1,045
N = 11.7 minutes
LESSON 6
Multiple rate problems are a little different.
Sometimes it may be helpful to draw a diagram because
you will be adding or subtracting the different rates
over a time period.
Here, let me show you:
An assembly line moves at a rate of 50 objects/
minute. If the rate goes up by 4 objects/min for 6
minutes, then slows down by 5 objects/min for 9
minutes before resuming normal speed, how many
objects have passed a worker stationed in the line in
the last hour?
When you look at this problem, you see several
different rates. It may come into perspective better when
you draw a diagram.
50 + 324 + 441 + 2,200 = 3,015 objects in one hour.
You have to really think hard on these.
EdWIN
Applied Mathematics • 97
LESSON 6
Example:
You open a savings account at Friendly Bank of America
with a $300 deposit. When you fill out the application,
the bank officer tells you the account earns 3% annual
interest, compounded monthly. You receive a letter from
the bank one month later announcing a rise in the interest
rate on savings accounts to 4%. How much interest will
you earn during the first two months?
When you look at interest for a year, you would earn 3%.
$300 × 3% = $9 for one year. But, you only earned this
for one month or
1
1
= $.75
of a year. $ 9 ×
12
12
At the end of the first month, you have $300.75.
You then earn 4%:
$300.75 × 4% ×
$12.03 ×
1
(for one month)
12
1
= $1.0025
12
When you add the two together, you earn $1.75. In this
problem, you had 2 rates that had to be considered.
Work the following rate problems. I suggest you
draw a diagram for the first one. As always, answers
will follow the exercise.
98 • Applied Mathematics
LESSON 6
EXERCISE – MULTIPLE RATE PROBLEMS
Instructions: Use proportions and/or diagrams to solve the following word problems.
1.
An airplane is flying at an altitude of 40,000 feet. Because of some
turbulence in the area, the pilot descends at a rate of 2,000 ft/min
for 3 minutes. He continues flying at this altitude until he decides
that the turbulence is over, then he begins an ascension at a rate of
1,500 ft/min and continues for 5.5 minutes. At what altitude is the
plane now flying?
2.
It takes 4.5 ounces of copper to make a copper pipe fitting. American
Pipe figures they produce an average of 25.4 fittings/week. How
much copper will it take to make fittings for the whole year?
3.
At the mouth of the Tennessee River, water flows at the rate of
1,600,000 cubic feet per second. How much water flows through
during one week?
Applied Mathematics • 99
LESSON 6
ANSWERS TO EXERCISE
1.
An airplane is flying at an altitude of 40,000 feet. Because of some
turbulence in the area, the pilot descends at a rate of 2,000 ft/min
for 3 minutes. He continues flying at this altitude until he decides
that the turbulence is over, then he begins an ascension at a rate of
1,500 ft/min and continues for 5.5 minutes. At what altitude is the
plane now flying?
Answer: 40,000 - (2,000 × 3) + (1,500 × 5.5)
40,000 - 6,000 + 8,250
42,250 ft
2.
It takes 4.5 ounces of copper to make a copper pipe fitting. American
Pipe figures they produce an average of 25.4 fittings/week. How
much copper will it take to make fittings for the whole year?
25.4(fittings)
Answer:
1(week)
=
N
52 (weeks)
N = 1,320.8
1,320.8 fittings made in 52 weeks or 1 year
1,320.8 × 4.5 (oz of copper) = 5,943.6 oz
100 • Applied Mathematics
LESSON 6
3.
At the mouth of the Tennessee River, water flows at the rate of
1,600,000 cubic feet per second. How much water flows through
during one week?
Answer: Calculate number of seconds in a week:
60 × 60 × 24 × 7 = 604,800
604,800 seconds in 1 week
Water flows at a rate of 1,600,000 cu ft/sec
Calculate cubic feet of water:
1,600,000 cu ft
1sec
=
N cu ft
604,800 sec
9.6768 × 1011 or
967,680,000,000 cu ft per week
Your calculator most likely displayed the answer in
“scientific notation.” The small number (exponent)
indicates the number of decimal places when using
scientific notation.
Applied Mathematics • 101
LESSON 7
REVIEW OF PERIMETER AND AREA
Perimeter
Perimeter is the measurement of the outside edges
of something (i.e., length of a fence, applied wallpaper
border, base boards in a room, etc.). Let’s review:
Rectangle – has 4 sides and each of 2 sides have equal
length
Triangle – has 3 sides which may or may not be of the
same length
Suppose the figure is a rectangle with length of 4 ft
and width of 2 ft.
Tip: Memorize how to find
perimeter and area.
A rectangle is the same length on opposite sides. This
means the rectangle measurements would be:
102 • Applied Mathematics
LESSON 7
Since perimeter is the measurement of the outside
distance, we can add these numbers.
4 ft + 4 ft + 2 ft + 2 ft = 12 ft
The perimeter is 12 ft. There is also a formula that you
could use
Perimeter = 2 times (length + width)
P = 2 (l + w)
Now, let’s go back to our rectangle.
P = 2 (l + w)
P = 2(4 ft + 2 ft) ← parentheses indicate
multiplication
P = 2 (6 ft)
P = 12 ft
You do not need to memorize these formulas to
take the ACT Work Keys Assessment. They will be on
the Formula Sheet distributed before the test. A copy
of the sheet is located in your references.
Let’s do another. Our rectangle has a length of 3
cm and a width of 1 cm.
P = 2 (l + w)
P = 2 (3 cm + 1 cm)
P = 2 (4 cm)
P = 8 cm
or you could add
P = 3 + 3 + 1 + 1 = 8 cm
Applied Mathematics • 103
LESSON 7
Finding the perimeter of a triangle is “similar” because
you add the lengths of the 3 sides of the triangle.
2 in
2 in
3 in
2 in + 2 in + 3 in = 7 in
The perimeter of this triangle is 7 inches.
Since the sides of a triangle do not have to be equal, we
let a different letter represent each side. The perimeter
of a triangle can be indicated as:
P=a+b+c
Area
Area is different from perimeter. Area measures the
surface of something (i.e., when you buy carpet you
need to measure the surface of the floor). When we
look at a rectangle,
area actually counts the squares inside the rectangle.
The area of the rectangle is 8 square feet.
4 ft
1 ft
2 ft
1 ft
1 ft
104 • Applied Mathematics
1 ft
1 ft
1 ft
LESSON 7
An easier way to calculate area is to use the formula
rather than count the number of square units.
Area = length times width
A=l×w
A=l×w
A = 4 ft × 2 ft
A = 8 sq ft
This measurement is in 2 dimensions and the unit of
measurement is often abbreviated as ft2.
Let’s try another problem. Find the area of a rectangle
6 m by 2 m.
A=l×w
A=6m×2m
A = 12 m2
Notice l × w is the same as w × l:
6 × 2 = 12 and 2 × 6 = 12
Applied Mathematics • 105
LESSON 7
Triangles require a different formula to calculate area.
The base is the bottom of the triangle. The height is
the perpendicular (vertical) line from the top point to
the base of the triangle.
1
A = (base × height)
2
1
A = ( 4 in × 8 in)
2
1
(32 in)
2
A = 16 in2
A=
Pop Quiz: A spool of nylon
manufactured at your plant
weighs 27
1
4
pounds. If a
customer orders 6 of these
spools of nylon, what will
be the shipping weight?
106 • Applied Mathematics
LESSON 7
Another problem might say: Find the area of the
triangle with a base of 12 cm and a height of 16 cm.
1
A = bh
2
1
A = (12 × 16)
2
1
A = (192)
2
A = 96 cm2
Let’s practice some perimeter and area problems now.
Applied Mathematics • 107
LESSON 7
EXERCISE – BASIC AREA AND PERIMETER PROBLEMS
Instructions: Solve the following problems using formulas to find perimeter and area.
1.
Carpeting is to be installed in a bedroom that is 14 feet long and
12
1
2
feet wide. What is the area of the room?
2.
You plan to put a fence around a lawn that is 200 feet by 380 feet
(rectangular in shape). How much fencing is needed?
3.
A customer wants to know how many tiles he will need to tile his
kitchen floor. His kitchen measures 14.5 feet by 15.2 feet. Each floor
tile is one square foot. How many tiles does he need? (Assume no
waste in cutting.)
108 • Applied Mathematics
LESSON 7
4.
The gable on a house needs to have the siding replaced. The gable
is 6 feet wide and 4 feet high. How many square feet of siding are
needed?
Applied Mathematics • 109
LESSON 7
ANSWERS TO EXERCISE
1.
Carpeting is to be installed in a bedroom that is 14 feet long and
12
1
2
feet wide. What is the area of the room?
Answer:
175 ft2
A=l×w
A = 14 ft × 12
2.
1
ft = 175 ft2
2
You plan to put a fence around a lawn that is 200 feet by 380 feet
(rectangular in shape). How much fencing is needed?
Answer:
1,160 ft
P = 2 (l + w)
P = 2(200 + 380)
P = 2 (580)
P = 1,160 ft
3.
A customer wants to know how many tiles he will need to tile his
kitchen floor. His kitchen measures 14.5 feet by 15.2 feet. Each floor
tile is one square foot. How many tiles does he need? (Assume no
waste in cutting.)
Answer:
221 tiles
A=l×w
A = 14.5 × 15.2 = 220.4 ft2
He needs 221 tiles
110 • Applied Mathematics
LESSON 7
4.
The gable on a house needs to have the siding replaced. The gable
is 6 feet wide and 4 feet high. How many square feet of siding are
needed?
Answer:
12 ft2
A=
1
bh
2
1
×6 ×4
2
A = 12 ft2
A=
Applied Mathematics • 111
LESSON 8
INTRODUCTION TO VOLUME
Lesson 8 will discuss how to find the volume of
rectangular solids. For this, you will need a formula.
The volume of a rectangular solid equals length times
width times height.
V=l×w×h
Take note of the formula
to calculate volume.
The answer will be given as a cubic measurement. Let
me show you:
Suppose you have a rectangular solid that is 6 cm by 4
cm by 3 cm.
6×4×3
6 × 4 = 24 then…
24 × 3 = 72 cm3 or 72 cubic centimeters
112 • Applied Mathematics
LESSON 8
Example:
A slab of concrete is to be poured into a frame that is 12 ft
by 6 ft by 6 in. The concrete is packaged by the cubic
yard. How much concrete is needed?
This problem asks for the answer in cubic yards. You will
need to convert your dimensions to yards. (If you can’t
remember how to do this, you may need to review
conversions through Level 4 or Level 5 of Applied
Mathematics).
12 ft = __ yd
6 ft = __ yd
6 in = __ yd
12 ft ×
6 ft ×
6 in ×
1 yd
3 ft
1 yd
3 ft
= 4 yd
= 2 yd
1 ft
12 in
×
1 yd
3 ft
=
6 1
= yd
36 6
Once you have done the conversion, it’s easy!
V = l×w×h
1
V = 4×2×
6
1
V = 1 cubic yards of concrete are needed
3
Applied Mathematics • 113
LESSON 8
Now, you practice. Just substitute the appropriate
measurements in the formula. The first one is tricky.
Try drawing a diagram before you peek at my answers.
Pop Quiz: It takes you 4 hours and 15 minutes to break down, clean, and
reassemble three machines. How long will it take you to clean 8 identical
machines?
114 • Applied Mathematics
LESSON 8
EXERCISE – PRACTICE FINDING VOLUME
Instructions: Find the volume of each rectangular solid to solve the following problems.
1.
An open tin box is to be constructed from a rectangular piece of tin
measuring 64 cm by 36 cm. The box will be made by cutting 6 cm
squares from each corner, then folding up the sides. What is the
volume of the box? It will help to make a sketch.
2.
Suppose a house has a rectangular pond in the front yard. Its
dimensions are 100 feet long, 75 feet wide, and 5 feet deep. Suppose
you want to put an “island water fall” in the middle using a square
column 10 feet by 10 feet by 5 feet deep. How many cubic feet of
water will be left in the pond?
Applied Mathematics • 115
LESSON 8
3.
Suppose a perfume bottle is in the shape of a cube. Each of its
dimensions are 2 cm. If the perfume cost is $65 per ounce, how
much will this bottle of perfume cost? (1 ounce = 8 cm3)
4.
A boxcar is 55 feet long, 12 feet wide, and 18 feet high. What is its
carrying capacity in cubic feet?
116 • Applied Mathematics
LESSON 8
ANSWERS TO EXERCISE
1.
An open tin box is to be constructed from a rectangular piece of tin
measuring 64 cm by 36 cm. The box will be made by cutting 6 cm
squares from each corner, then folding up the sides. What is the
volume of the box? It will help to make a sketch.
Answer: 64 - 12 = 52 length of box
36 - 12 = 24 width of box
52 × 24 × 6 = 7,488 cm3
(This one is easy to understand if you take a sheet of
paper and actually cut squares off the corners).
Applied Mathematics • 117
LESSON 8
2.
Suppose a house has a rectangular pond in the front yard. Its
dimensions are 100 feet long, 75 feet wide and 5 feet deep. Suppose
you want to put an “island water fall” in the middle using a square
column 10 feet by 10 feet by 5 feet deep. How many cubic feet of
water will be left in the pond?
Answer: 100 × 75 × 5 = 37,500 ft3 volume of pond
10 × 10 × 5 = 500 ft3 volume of column
37,500 - 500 = 37,000 cubic feet of water left
118 • Applied Mathematics
LESSON 8
3.
Suppose a perfume bottle is in the shape of a square cube. Each of
its dimensions are 2 cm. If the perfume cost is $65 per ounce, how
much will this bottle of perfume cost? (1 ounce = 8 cm3)
Answer: 2 × 2 × 2 = 8 cm3
8 cm 3 ×
1 oz
8 cm 3
= 1 oz
$65 × 1 = $65
4.
A boxcar is 55 feet long, 12 feet wide, and 18 feet high. What is its
carrying capacity in cubic feet?
Answer: V = l × w × h
V = 55 × 12 × 18
V = 11,880 ft3 or 11,880 cubic feet
Applied Mathematics • 119
LESSON 9
APPLICATIONS OF MULTISTEP WORD
PROBLEMS
The first part of Lesson 9 involves sorting through
your information and performing several operations.
Diagrams are often helpful when solving multistep
problems. Let me show you an example:
You usually work from 7:00 a.m. - 3:00 p.m. five days a
week, Saturday through Wednesday. If you work any
overtime, you are paid time-and-a-half. Your normal hourly
pay rate is $7.74/hour. You are also compensated double
time for any special projects you are working on outside
of your normal shift. During the week beginning Saturday,
Oct. 15, you went to work at 7:00 a.m. on Saturday
through Friday. You left at 3:00 p.m. on Wednesday,
Thursday, and Friday, but you stayed until 4:00 p.m. on
the other days. You spent the whole day Thursday working
with a co-worker planning a special car show which your
company is sponsoring to benefit a local school for disabled
children. How much should you be paid for the week?
Normal week
Saturday – Wednesday 7:00 a.m. – 3:00 p.m.
$7.74 per hour
Overtime pay
$11.61 per hour
Special project pay
$15.48 per hour (outside normal hours)
120 • Applied Mathematics
LESSON 9
Paycheck for the week = $572.76 (before deductions)
Now, it’s time for you to practice sorting through
the information. This is probably something you have
either actually done in the workplace or will do
eventually. Good Luck!
Applied Mathematics • 121
LESSON 9
EXERCISE – MULTISTEP PROBLEMS
Instructions: Solve the following word problems. Use diagrams to help organize the
information.
1.
The president of your company will be in town next month for 5
days. Before the president comes to visit, the lobby area, illustrated
below, must be remodeled. Gathering prices for the carpeting is
your responsibility. The carpet the decorator has chosen costs $8.99
per square yard. You know an installer who charges $2.50 per square
yard, but he gives a 15% discount off jobs that are more than 100
square yards. The padding costs $2.99 per square yard. Find the
cost of having new carpet and padding installed.
122 • Applied Mathematics
LESSON 9
2.
You bought a cellular phone and have signed a 1-year contract. In
your contract, you agreed to a monthly service charge of $12.50
plus phone calls. The charge for calls made during the peak hours
of 7 a.m. - 7 p.m. is $.50 per call and the charge for calls made
during off-peak hours is $.30 per call. During the month of March,
you made 14 calls to your wife on your way home from work (after
7 p.m.) and you made 43 calls during regular business hours. Your
monthly bill will be how much?
Applied Mathematics • 123
LESSON 9
ANSWERS TO EXERCISES
1.
The president of your company will be in town next month for 5
days. Before the president comes to visit, the lobby area, illustrated
below, must be remodeled. Gathering prices for the carpeting is
your responsibility. The carpet the decorator has chosen costs $8.99
per square yard. You know an installer who charges $2.50 per square
yard, but he gives a 15% discount off jobs that are more than 100
square yards. The padding costs $2.99 per square yard. Find the
cost of having new carpet and padding installed.
Answer:
A = 34 × 11 = 374 sq ft
B = 25 × 13 = 325 sq ft
374 + 325 = 699 sq ft total area
Convert to yards (may be converted prior to calculations)
699 ÷ 9 = 77.7 sq yd
Installation
$2.50 × 77.7 = $194.25 installation fee
(order does not qualify for discount)
Carpet
$8.99 × 77.7 = $698.52
Padding
$2.99 × 77.7 = $232.32
Total
$194.25 + $698.52 + $232.32 = $1,125.09
124 • Applied Mathematics
LESSON 9
2.
You bought a cellular phone and have signed a 1-year contract. In
your contract, you agreed to a monthly service charge of $12.50
plus phone calls. The charge for calls made during the peak hours
of 7 a.m. - 7 p.m. is $.50 per call and the charge for calls made
during off-peak hours is $.30 per call. During the month of March,
you made 14 calls to your wife on your way home from work (after
7 p.m.) and you made 43 calls during regular business hours. Your
monthly bill will be how much?
Answer: $38.20
$12.50 (base charge)
$4.20 (14 calls × .30 = $4.20)
$ 21.50 (43 calls × .50 = $21.50)
$12.50 + $4.20 + $21.50 = $38.20 Total bill
Applied Mathematics • 125
LESSON 9
Decision problems are a big part of life. There are
many things that you must decide every day. Sometimes
you have to do calculations before you make a decision.
In these problems, you should already know how to
do the calculations. You must decide which product is
a better buy or which is a better rate.
Take a look at the following problems and try to
make the best decision.
126 • Applied Mathematics
LESSON 9
EXERCISE – DECISION PROBLEMS
Instructions: Solve the following decision problems.
1.
You received a $5,000 bonus last year and are considering investing
it. Your financial planner suggests that you invest 40% in a stock
paying 7% annual dividends and the rest in a money market fund
that earns 4.5% annual interest. Your banker suggests you invest
the entire amount in a certificate of deposit earning 5% annual rate
compounded every 6 months. Which advice should you follow in
order to get the highest yield from your investment?
2.
You are considering changing jobs and there are two positions you
must decide between. One job has an hourly pay rate of $6.50/hour
plus 6.25% commission on every sale. (Assume you work 40 hours
per week, 50 weeks per year and have average sales of $5,400 per
week). The other job pays a yearly salary of $25,000 plus a 4% bonus
at the end of the year. Basing your decision only on earnings, which
job should you choose?
Applied Mathematics • 127
LESSON 9
3.
You are planning refreshments for your son’s 4th birthday party.
The smallest cake that will feed your group serves 24 people and
costs $13.50 or you could order cupcakes at $.75 per cupcake. For
beverages, you must decide between sodas at $.30 per can or a
punch that serves 25 and costs $5.00 to make. If you are expecting
20 children, which would be the most cost efficient?
Happy Birthday
128 • Applied Mathematics
LESSON 9
ANSWERS TO EXERCISE
1.
You received a $5,000 bonus last year and are considering investing
it. Your financial planner suggests that you invest 40% in a stock
paying 7% annual dividends and the rest in a money market fund
that earns 4.5% annual interest. Your banker suggests you invest
the entire amount in a certificate of deposit earning 5% annual rate
and compounded every 6 months. Which advice should you follow
in order to get the highest yield from your investment?
Answer: Option 1 – Financial Planner
5,000 × 40% = $2,000
2,000 × 7% = $140 return on stock dividends
$3,000 (remainder of bonus) × 4.5% = $135 return from
money market
So, $140 + $135 = $275 total yield
Option 2 – Banker
$5,000 × 5% = $250 per year
1
Earned
of $250 in 6 months (5% is annual rate)
2
1
× $250 = $125
2
$5,000 + $125 = $5,125 reinvested in second 6 months
$5,125 × 5 % = $256.25
1
× $256.25 = $128.13
2
Add $125 + $128.13 = $253.13 total yield
Option 1 – $275
Option 2 – $253.13
The financial planner’s advice was better.
Applied Mathematics • 129
LESSON 9
2.
You are considering changing jobs and there are two positions you
must decide between. One job has an hourly pay rate of $6.50/hour
plus 6.25% commission on every sale. (Assume you work 40 hours
per week, 50 weeks per year and have average sales of $5,400 per
week.) The other job pays a yearly salary of $25,000 plus a 4% bonus
at the end of the year. Basing your decision only on earnings, which
job should you choose?
Answer:
Option 1
$6.50 × 40 (hours) = $260 per week
$260 × 50 (weeks) = $13,000 salary
$5,400 × 50 (weeks) = $270,000
$270,000 × 6.25% (commission)
= $16,875 (commission)
Total from Option 1
$13,000 + $16,875 = $29,875
Option 2
$25,000 base
4% of $25,000 = $1,000 bonus
Total from Option 2
$25,000 + $1,000 = $26,000
Option 1 – $29,875
Option 2 – $26,000
Based on earnings, Option 1 is a better offer.
130 • Applied Mathematics
LESSON 9
3.
You are planning refreshments for your son’s 4th birthday party.
The smallest cake that will feed your group serves 24 people and
costs $13.50 or you could order cupcakes at $.75 per cupcake. For
beverages, you must decide between sodas at $.30 per can or a
punch that serves 25 and costs $5.00 to make. If you are expecting
20 children, which would be the most cost efficient?
Answer: Food:
$13.50 cake
$.75 × 20 = $15.00 cupcakes
The cake would be the most cost effective decision.
Drink:
$5.00 punch
$.30 × 20 = $6.00 sodas
The punch costs less.
Applied Mathematics • 131
LESSON 9
The very last part of this level may seem trivial,
but in fact, it is very important. We need to know how
to search for mistakes in problems. This is significant
to industry because it is important to catch mistakes
before the final product is complete. Understanding
where a mistake was made can help prevent its
reoccurrence in the future. You should develop a
checklist which includes common errors to look for.
Your checklist could start with the following questions:
Get busy and make your
checklist!
•
Were calculations done using numbers all
converted to the same units?
•
Were all decimals placed correctly?
•
Were operations (e.g., addition, subtraction)
performed correctly or, if using a calculator,
were numbers entered correctly?
•
Were operations performed in the correct order
(e.g., a discount applies to only one of several
items a customer is purchasing and should be
taken off before all items are totaled)?
•
Was the correct formula used and/or were
values substituted appropriately (e.g., radius
instead of diameter)?
•
If fractions were used, were common
denominators found or inversions done when
necessary?
•
Were conversions of percents done correctly?
•
Were operations on mixed units, or
denominate numbers, done correctly?
Adapted from WorkKeys® Targets for Instruction: Applied Mathematics,
© 1997 by ACT, Inc.
132 • Applied Mathematics
LESSON 9
Let’s look at a problem.
Example:
You want to enclose a round sand box 3 feet wide with
edging. How much edging do you need?
Given answer: 28.26 ft
You need to enclose the outside edges. This will be
circumference. You learned previously that C = πd. The
diameter is 3 ft.
C = π(3) = 9.42 ft using 3.14 as an approximation for π.
The given answer is 28.26 ft. This answer squared 3 ft.
π(3)2 (finding area if the radius were 3)
This would be very important if working on a budget.
You would have just bought three times the material you
needed.
Now, see if you can find the mistakes in the
following problems. Think about what problems the
mistakes might cause.
Applied Mathematics • 133
LESSON 9
EXERCISE – FIND THE MISTAKES
Instructions: Determine if each problem is worked correctly and, if not, find the mistake
that was made.
1.
It was estimated that it would take 440 minutes to complete a job,
but it actually took 380 minutes. How many hours did it actually
take to complete the job?
ANSWER: 3 hours 80 minutes
2.
What percent of 50 is 80?
ANSWER: 160%
134 • Applied Mathematics
LESSON 9
3.
A jacket that originally sold for $40 was marked down to $32. What
was the percent of the discount?
ANSWER: 25%
4.
A solution of water and bleach is to be mixed in a 4:1 ratio. If 4 cups
of the mixture is desired, how much of each do you need to use?
ANSWER: 3.2 cups of water and 0.8 cups of bleach
Applied Mathematics • 135
LESSON 9
ANSWERS TO EXERCISE
1.
It was estimated that it would take 440 minutes to complete a job,
but it actually took 380 minutes. How many hours did it actually
take to complete the job?
ANSWER: 3 hours 80 minutes
Answer:
380 min ×
1 hr
= 6.3 hr
60 min
The mistake was the assumption 100 minutes in an hour.
2.
What percent of 50 is 80?
ANSWER: 160%
Answer:
N% × 50 = 80
80
50
N% = 1.6
N% = 160%
N% =
The given answer is correct.
3.
A jacket that originally sold for $40 was marked down to $32. What
was the percent of the discount?
ANSWER: 25%
Answer:
Discount: $40 - $32 = $8
$8 decrease
$40 original
.2 = 20% discount
The mistake was in using $32 instead of $40 as the
original price.
136 • Applied Mathematics
LESSON 9
4.
A solution of water and bleach is to be mixed in a 4:1 ratio. If 4 cups
of the mixture is desired, how much of each do you need to use?
ANSWER: 3.2 cups of water and 0.8 cups of bleach
Answer: 4:1 ratio means 4 cups of water to 1 cup bleach
This ratio represents a total of 5 cups.
Water:
4 c water
5 c total
=
x c water
4 c total
5x = 16
x=3
1
or 3.2 cups water
5
Bleach:
1 c bleach
5 c total
=
x c bleach
4 c total
5x = 4
x=
4
or .8 cups bleach
5
The given answer is correct.
Applied Mathematics • 137
LESSON 10
You have just completed the last lesson of this level.
Take a break before you complete the following Posttest.
See how much knowledge you have retained in
preparation for the ACT Work Keys Assessment.
I trust that you did well throughout Level 6. Let’s
see how much you remember by taking the Posttest on
the following pages. Good luck and don’t peek at the
answers!
138 • Applied Mathematics
POSTTEST
EXERCISE – POSTTEST
Instructions: Perform the operations to solve the following problems.
1.
What is 20% of 50?
2.
18 is what percent of 90?
3.
Sheri wrote the following entries in her checkbook this week:
-$13.52, -$15.88, +$500, +$2,516. What is the average amount of
checks written?
4.
You must mark prices for a rack of sweaters which were originally
marked $36.99. If they are on sale for 35% off, what will be the sale
price?
Applied Mathematics • 139
POSTTEST
5.
What is -5 + (-2)?
6.
What is -0.9 × -0.2?
7.
What is the carrying capacity of a boxcar 55 ft by 12 ft by 18 ft in
cubic yards?
8.
What is
3 7
+ ?
4 16
9.
What is
3 9
÷ ?
8 10
140 • Applied Mathematics
POSTTEST
10. What is −
1
1
+4 ?
2
3
11. Once a week 40 employees where you work have pizza delivered
for lunch. The current caterer, Papa’s Pizza, has a delivery fee of
$20 plus a price of $3.00 per each personal size pizza. A new pizzeria,
Mama Mia’s, will provide personal size pizzas for $2.50 each, but a
$30 delivery fee is charged. Considering costs, should you change
caterers?
12. Determine the cost of cement needed for a sidewalk that is 4 inches
thick, 3 feet wide, and 40 feet long. The cement is sold for $76 per
cubic yard. (Calculate problem using fractions to avoid rounding
errors caused by repeating decimals.)
Applied Mathematics • 141
POSTTEST
13. You are preparing to fertilize your front lawn which is a rectangular
shape measuring 90
1
2
yards by 35
1
2
yards. A 25 lb bag of fertilizer
covers approximately 5,000 square feet. How many bags of fertilizer
need to be purchased?
14. The health food store that you manage employs 7 clerks. Individual
hourly wages are $6.10, $6.25, $6.45, $6.75, $7.00, $7.25, and $8.50.
If each employee has 2 weeks unpaid vacation and works 8 hours a
day, 5 days per week and is paid for all holidays, what is the total
amount of annual payroll for the clerks?
142 • Applied Mathematics
POSTTEST
15. On your first day at work at the Prissy Puppy Grooming Shop, you
groomed 3 dogs between 7:45 a.m. and 2:30 p.m., taking breaks
that totaled 45 minutes. Your supervisor advises that you will be
scheduling your own appointments as customers call to have their
pets groomed. Approximately how long should you allow yourself
for a grooming appointment?
16. An assembly line moves at a rate of 25 objects/min. If the rate goes
up by 2 objects/min for 6 minutes, then slows down by 4 objects/
min for 9 minutes before resuming normal speed, how many objects
have passed a worker stationed on the line during the last one-half
hour?
Applied Mathematics • 143
POSTTEST
17. You work with a group of amateur actors designing and building
sets. You need a black curtain 10 feet high by 30
1
2
feet across to
cover the back of the stage. The fabric you select is 6 feet wide. You
decide to sew two pieces together, allowing 2 inches per piece for
the center seam, to attain the height of the curtain. If you allow 6
inches per side to be turned under and hemmed, how many yards
of fabric do you need to make the curtain? (See diagram.)
18. Find the mistake if the answer given to the following problem is:
Carbohydrates
1,000 calories
Fat
1,325 calories
Protein
625 calories
A doctor prescribes a diet of 2,500 calories per day for a patient.
She specifies that 40% of the calories must be carbohydrates, 35%
must be fat, and 25% must be protein. Calculate the number of
calories prescribed for the patient in each category.
144 • Applied Mathematics
POSTTEST
19. The production line where you work can assemble alarm clocks at
a rate of 15 every 45 minutes. How long will it take to assemble 100
alarm clocks at this rate?
20. You work in the music shop at the mall. A customer purchased 2
CDs for $14.95 each, a cassette tape for $7.99, and a 5-pack of blank
cassettes for $5.99. He gave you a $50 bill and you gave him $21.07
change. How much, if at all, did you overcharge or undercharge the
customer?
Applied Mathematics • 145
POSTTEST
ANSWERS TO EXERCISE
1.
What is 20% of 50?
Answer: N = 20% × 50
N = 10
2.
18 is what percent of 90?
Answer: 18 = N% × 90
18
= N%
90
.2 = N%
20% = N%
3.
Sheri wrote the following entries in her checkbook this week:
-$13.52, -$15.88, +$500, +$2,516. What is the average amount of
checks written?
Answer:
4.
$13.52 + $15.88 = $29.40
Average of 2 checks: $29.40 ÷ 2 = $14.70
(The 2 deposits are extraneous information in this
problem.)
You must mark prices for a rack of sweaters which were originally
marked $36.99. If they are on sale for 35% off, what will be the sale
price?
Answer: 35% of $36.99 is N
$12.95 is the discount
$36.99 - $12.95 = $24.04 sale price
5.
What is -5 + (-2)?
Answer: -7
146 • Applied Mathematics
POSTTEST
6.
What is -0.9 × -0.2?
Answer: .18
7.
What is the carrying capacity of a boxcar 55 ft by 12 ft by 18 ft in
cubic yards?
Answer:
55 ft ×
12 ft ×
18 ft ×
1 yd
3 ft
1 yd
3 ft
1 yd
3 ft
= 18.333... yd
= 4 yd
= 6 yd
V=l×w×h
V = 18.333... × 4 × 6
V = 440 cubic yards
This problem may be worked in feet then converted to
yards.
V = 55 × 12 × 18
V = 11,880 cu ft
V = 11,880 cu ft ×
1 cu yd
27 cu ft
= 440 cu yd
Applied Mathematics • 147
POSTTEST
8.
What is
3 7
+ ?
4 16
19
3
or 1
16
16
Answer:
9.
What is
3 9
÷ ?
8 10
Answer:
10. What is -
5
12
1
1
+4 ?
2
3
Answer: 3
5
6
148 • Applied Mathematics
POSTTEST
11. Once a week 40 employees where you work have pizza delivered
for lunch. The current caterer, Papa’s Pizza, has a delivery fee of
$20 plus a price of $3.00 per each personal size pizza. A new pizzeria,
Mama Mia’s, will provide personal size pizzas for $2.50 each, but a
$30 delivery fee is charged. Considering costs, should you change
caterers?
Answer: Find the cost per employee using Papa’s Pizza.
40 × 3.00 = $120.00 plus the delivery fee $20
which totals $140.00
140.00
= $3.50 per person
40
Find the cost per employee using Mama Mia’s.
40 × 2.50 = $100 plus delivery fee $30
which totals $130.00
130.00
= $3.25 per person
40
Yes, each employee would save $3.50 - $3.25 = $0.25
per meal by changing caterers.
Applied Mathematics • 149
POSTTEST
12. Determine the cost of cement needed for a sidewalk that is 4 inches
thick, 3 feet wide, and 40 feet long. The cement is sold for $76 per
cubic yard. (Calculate problem using fractions to avoid rounding
errors caused by repeating decimals.)
Answer: Convert units to yards.
1
1 ft
in = ft
3
12 in
4 in ×
1
1
1 yd
ft ×
= yd thickness (height)
3
9
3 ft
3 ft ×
1 yd
= 1 yd width
3 ft
40 ft ×
40
1 yd
=
yd length
3
3 ft
V=l×w×h
V =
1
40 40
13
×1 ×
=
=1
cu yd
9
3
27
27
40
= $112.59
27
It will cost $112.59 for the cement.
$76 ×
150 • Applied Mathematics
POSTTEST
13. You are preparing to fertilize your front lawn which is a rectangular
shape measuring 90
1
2
yards by 35
1
2
yards. A 25 lb bag of fertilizer
covers approximately 5,000 square feet. How many bags of fertilizer
need to be purchased?
Answer: Convert units to decimals
1
35 = 35.5
2
90
1
= 90.5
2
Find area of front lawn
35.5 × 90.5 = 3,212.75 sq yd
Convert yards to feet:
9 sq ft
3,212.75 sq yd ×
1sq yd
= 3.212.75 × 9
= 28,914.75 sq ft
28,914.75 ÷ 5,000 (area covered by 1 bag) = 5.78
The fact that each bag weighs 25 lb is not relevant to
calculating the number of bags needed.
It will take 5.78 bags of fertilizer to cover the lawn, so 6
bags will need to be purchased.
Applied Mathematics • 151
POSTTEST
14. The health food store that you manage employs 7 clerks. Individual
hourly wages are $6.10, $6.25, $6.45, $6.75, $7.00, $7.25, and $8.50.
If each employee has 2 weeks unpaid vacation and works 8 hours a
day, 5 days per week and is paid for all holidays, what is the total
amount of annual payroll for the clerks?
Answer: Calculate the number of working hours per year (52
weeks in a year minus 2 weeks vacation = 50)
8 (hours) × 5 (days) × 50 (weeks)
= 2,000 work hours per year
The annual payroll includes all wages, so each
employee’s wages must be calculated.
6.10 × 2,000 = $12,200
6.25 × 2,000 = $12,500
6.45 × 2,000 = $12,900
6.75 × 2,000 = $13,500
7.00 × 2,000 = $14,000
7.25 × 2,000 = $14,500
8.50 × 2,000 = $17,000
Total = $96,600
Total amount of the annual payroll is $96,600.
152 • Applied Mathematics
POSTTEST
15. On your first day at work at the Prissy Puppy Grooming Shop, you
groomed 3 dogs between 7:45 a.m. and 2:30 p.m., taking breaks
that totaled 45 minutes. Your supervisor advises that you will be
scheduling your own appointments as customers call to have their
pets groomed. Approximately how long should you allow yourself
for a grooming appointment?
Answer: Because this problem involves a.m. and p.m., we will
break the calculations into 2 parts: the time from 7:45
until noon and the time from noon until 2:30.
Find the difference from 7:45 a.m. until noon:
12:00 (noon may be converted to 11 hr 60 min)
11 hr 60 min
-7 hr 45 min
4 hr 15 min
The time from noon until 2:30 is 2 hr and 30 min.
Combining morning and afternoon times:
4 hr 15 min
+2 hr 30 min
6 hr 45 min
Subtract 45 minutes for breaks to find it took 6 hours for
3 dogs to be groomed.
6 hr
3 dogs
=
N hr
1 dog
3N = 6
N = 2 hours for each dog
Applied Mathematics • 153
POSTTEST
16. An assembly line moves at a rate of 25 objects/min. If the rate goes
up by 2 objects/min for 6 minutes, then slows down by 4 objects/
min for 9 minutes before resuming normal speed, how many objects
have passed a worker stationed on the line during the last one-half
hour?
Answer:
Total objects for 30 minutes:
25 + 162 + 207 + 350 = 744 objects
154 • Applied Mathematics
POSTTEST
17. You work with a group of amateur actors designing and building
sets. You need a black curtain 10 feet high by 30
1
2
feet across to
cover the back of the stage. The fabric you select is 6 feet wide. You
decide to sew two pieces together, allowing 2 inches for the center
seam, to attain the height of the curtain. If you allow 6 inches per
side to be turned under and hemmed, how many yards of fabric do
you need to make the curtain? (See diagram.)
Answer: Convert 30
1
1
ft to in (30 = 30.5)
2
2
12 in
30.5 ft ×
1 ft
= 366 inches
Calculate total length of curtain (fabric) pieces:
366 (curtain) + 6 (side) + 6 (side) = 378 inches
You need 2 panels this size to attain the length needed:
378 × 2 = 756 inches
Convert result to yards:
1 ft
756 in ×
63 ft ×
12 in
= 63 ft
1 yd
= 21 yd
3 ft
I would buy 21 yards of fabric.
Applied Mathematics • 155
POSTTEST
18. Find the mistake if the answer given to the following problem is:
Carbohydrates
1,000 calories
Fat
1,325 calories
Protein
625 calories
A doctor prescribes a diet of 2,500 calories per day for a patient.
She specifies that 40% of the calories must be carbohydrates, 35%
must be fat, and 25% must be protein. Calculate the number of
calories prescribed for the patient in each category.
Answer: The number of calories calculated for fat was incorrect;
it should be obvious since 40% is greater than 35% and
the given answer indicates a higher number of calories
from fat rather than carbohydrates. The mistake was in
entering 53% rather than 35% for the percentage of fat.
Carbohydrates
Fat
Protein
1,000 calories
875 calories
625 calories
19. The production line where you work can assemble alarm clocks at
a rate of 15 every 45 minutes. How long will it take to assemble 100
alarm clocks at this rate?
Answer: Use a proportion to calculate the time needed for
100 clocks.
15 clocks 100 clocks
=
45 min
N min
15N = 4,500
N = 300 minutes
(continued)
156 • Applied Mathematics
POSTTEST
Convert 300 minutes to hours:
1 hr
300 min ×
60 min
= 5 hr
It takes 5 hours to assemble 100 alarm clocks.
(You may work this problem in hours or minutes.)
20. You work in the music shop at the mall. A customer purchased 2
CDs for $14.95 each, a cassette tape for $7.99, and a 5-pack of blank
cassettes for $5.99. He gave you a $50 bill and you gave him $21.07
change. How much, if at all, did you overcharge or undercharge the
customer?
Answer: Total the items:
(2 × 14.95) + 7.99 + 5.99
29.90 + 7.99 + 5.99 = $43.88
$50 - $43.88 = $6.12 correct change
You undercharged the customer by ($21.07 - $6.12)
= $14.95
It looks like you only charged him for one CD.
Applied Mathematics • 157
CALCULATING YOUR SCORE
Calculate your score counting the number of questions you answered correctly. Divide
the number of your correct answers by 20. Change the decimal answer to a percent by
moving the decimal two places to the right.
158 • Applied Mathematics
SUMMARY
Well, how did you do on the Posttest? If you scored
95% or higher, you have a reasonable chance to pass
Level 6 of the ACT WorkKeys® Applied Mathematics
assessment. Remember the basic steps for solving
mathematics problems. Take your time and think about
each question and you will do fine. You may want to
complete Level 7, the final course of Applied
Mathematics, with me before you take the assessment.
Hope to see you there.
Now don’t be discouraged if you scored below 95%.
There is a lot of information to remember. You know
you can do it! Your enhanced work skills will pay off in
the long run.
Take time to review the Workplace Problem Solving
Glossary and Test-Taking Tips provided at the end of
this workbook. Good luck improving your work skills
and attaining your goals!
You should be proud of
your progress.
Applied Mathematics • 159
REFERENCE
WORKPLACE PROBLEM SOLVING GLOSSARY
The following is a partial list of words that has been compiled for you to review before
taking the ACT WorkKeys® Applied Mathematics assessment. The assessment consists of
approximately 33 application (word) problems that focus on realistic workplace situations.
It is important that you are familiar with common workplace vocabulary so that you
may interpret and determine how to solve the problems.
Annual - per year
Asset - anything of value
Budget - estimate of income and expenses
Capital - money, equipment, or property used in a business by a person or corporation
Capital gain (loss) - difference between what a capital asset costs and what it sells for
Commission - an agent’s fee; payment based on a percentage of sales
Contract - a binding agreement
Convert - to change to another form
Deductions - subtractions
Denominate number - numbers with units i.e., 5 feet, 10 seconds, 2 pounds
Depreciation - lessening in value
Difference - answer to a subtraction
Discount - reduction from a regular price
Dividend - money a corporation pays to its stockholders
Expense - cost
160 • Applied Mathematics
REFERENCE
Fare - price of transportation
Fee - a fixed payment based on a particular job
Fiscal year - 12-month period a corporation uses for bookkeeping purposes
Gross pay - amount of money earned
Gross profit - gross pay less immediate cost of production; difference in sales price of
item or service and expenses attributed directly to it
Interest - payment for use of money; fee charged for lending money
Interest rate - rate percent per unit of time i.e., 7% per year
Liquid Asset - current cash or items easily converted to cash
Markup - price increase
Measure - a unit specified by a scale, such as an inch
Net pay - take-home pay; amount of money received after deductions
Net profit (income) - actual profit made on a sale, transaction, etc., after deducting all
costs from gross receipts
Overtime - payment for work done in addition to regular hours
Per - for each
Percent off - fraction of the original price that is saved when an item is bought on sale
Product - answer to a multiplication problem
Profit - income after all expenses are paid
Proportion - an equation of 2 ratios that are equal
Applied Mathematics • 161
REFERENCE
Quotient - answer to a division problem
Rate - a ratio or comparison of 2 different kinds of measures
Ratio - a comparison of 2 numbers expressed as a fraction, in colon form, or with the
word “to”
Regular price - price of an item not on sale or not discounted
Return rate - percentage of interest or dividends earned on money that is invested
Revenue - amount of money a company took in ( interest, sales, services, rents, etc.)
Salary - a fixed rate of payment for services on a regular basis
Sale price - price of an item that has been discounted or marked down
Sum - answer to an addition problem
Yield - amount of interest or dividends an investment earns
162 • Applied Mathematics
REFERENCE
EDWIN’S TEST-TAKING TIPS
Preparing for the test . . .
Complete appropriate levels of the WIN Instruction Solution self-study courses. Practice
problems until you begin to feel comfortable working the word problems.
Get a good night’s rest the night before the test and eat a good breakfast on test day.
Your body (specifically your mind) works better when you take good care of it.
You should take the following items with you when you take the ACT WorkKeys®
Applied Mathematics assessment: (1) pencils; pens are not allowed to be used on the
test; it is a good idea to have more than one pencil since the test is timed and you do
not want to waste time sharpening a broken pencil lead; and (2) your calculator; be
sure your batteries are strong if you do not have a solar-powered calculator and that
your calculator is working properly.
Allow adequate time to arrive at the test site. Being in a rush or arriving late will likely
upset your concentration when you actually take the test.
About the test . . .
The test is comprised of approximately 33 multiple-choice questions. All test questions
are in the form of word problems which are applicable to the workplace. You will not
be penalized for wrong answers, so it is better to guess than leave blanks. You will
have 45 minutes to complete the test.
The test administrator will provide a Formula Sheet exactly like the one provided in
this workbook. You will not be allowed to use scratch paper, but there is room in your
assessment booklet to work the problems.
Applied Mathematics • 163
REFERENCE
During the test . . .
Listen to instructions carefully and read the test booklet directions. Do not hesitate
to ask the administrator questions if you do not understand what to do.
Pace yourself since this is a timed test. The administrator will let you know when you
have 5 minutes left and again when you have 1 minute remaining. Work as quickly as
possible, but be especially careful as you enter numbers into your calculator.
If a problem seems too difficult when you read it, skip over it (temporarily) and move
on to an easier problem. Be sure to put your answers in the right place. Sometimes
skipping problems can cause you to get on the wrong line, so be careful. You might
want to make a mark in the margin of the test, so that you will remember to go back
to any skipped problems.
Since this is a multiple-choice test, you have an advantage answering problems that
are giving you trouble. Try to eliminate any unreasonable answers and make an
educated guess from the answers you have left.
If the administrator indicates you have one minute remaining and you have some
unanswered questions, be sure to fill in an answer for every problem. Your guess is
better than no answer at all!
If you answer all of the test questions before time is called, use the extra time to check
your answers. It is easy to hit the wrong key on a calculator or place an answer on the
wrong line when you are nervous. Look to see that you have not accidentally omitted
any answers.
164 • Applied Mathematics
REFERENCE
Dealing with math anxiety . . .
Being prepared is one of the best ways to reduce math or test anxiety. Study the list of
key words for solving word problems. If your problem does not include any key
words, see if you can restate the problem using your key words. Feeling like you know
several ways to try to solve problems increases your confidence and reduces anxiety.
Do not think negatively about the test. The story about the “little engine that could”
is true. You must, “think you can, think you can, think you can.” If you prepare
yourself by studying problem solving strategies, there is no reason why you cannot be
successful.
Do not expect yourself to know how to solve every problem. Do not expect to know
immediately how to work word problems when you read them. Everyone has to read
and reread problems when they are solving word problems. So, don’t get discouraged;
be persistent.
Prior to the test, close your eyes, take several deep breaths, and think of a relaxing
place or a favorite activity. Visualize this setting for a minute or two before the test is
administered.
During the test if you find yourself tense and unable to think, try the following
relaxation technique:
1.
2.
3.
4.
5.
Put feet on floor.
Grab under your chair with your hands. (hope there are no surprises!)
Push down with your feet and up on your chair at the same time - hold for 5
seconds.
Relax 5 seconds (especially try to relax your neck and shoulders).
Repeat a couple of times as needed, but do not spend the entire 45 minutes of
the test trying to relax!
Studying with a partner is another way to overcome math anxiety. Encouragement
from each other helps to increase your confidence.
Applied Mathematics • 165
REFERENCE
FORMULA SHEET
(≈ indicates estimate, not equal)
166 • Applied Mathematics
ANSWERS TO POP QUIZ QUESTIONS
POP QUIZ QUESTION ANSWER KEY
1. Page 84 – Option 1
Double prints whole roll = 36 × .45 = $16.20
$16.20 × 15% (discount) = $2.43
$16.20 - $2.43 = $13.77
Option 2
Single = 36 × .32 = $11.52
$11.52 + (9 × .51) = $11.52 (single)
$11.52 (single) + $4.59 (reprints) = $16.11
It’s cheaper to get doubles of the whole roll but only with
the 15% discount.
2. Page 106 – 163
1
2
pounds
3. Page 114 – Convert hr to min:
4 hr ×
60 min
1hr
= 240 min
240 (4 hr) + 15 = 255 min
Use proportion to calculate:
255 min
3 machines
=
x min
8 machines
3x = 255 × 8
3x = 2,040
x = 680 minutes
Convert answer back to hours:
680 min ×
680
60
1 hr
60 min
=
680
60
= 11.3 hours or 11 hours and 20 minutes
Applied Mathematics • 167
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